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Frontiers of Structural and Civil Engineering

Front. Struct. Civ. Eng.    2019, Vol. 13 Issue (1) : 38-48
Empirical models and design codes in prediction of modulus of elasticity of concrete
Centre for Built Infrastructure Research, University of Technology Sydney, Sydney, Australia
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Modulus of Elasticity (MOE) is a key parameter in reinforced concrete design. It represents the stress-strain relationship in the elastic range and is used in the prediction of concrete structures. Out of range estimation of MOE in the existing codes of practice strongly affect the design and performance of the concrete structures. This study includes: (a) evaluation and comparison of the existing analytical models to estimating the MOE in normal strength concrete, and (b) proposing and verifying a new model. In addition, a wide range of experimental databases and empirical models to estimate the MOE from compressive strength and density of concrete are evaluated to verification of the proposed model. The results show underestimation of MOE of conventional concrete in majority of the existing models. Also, considering the consistency between density and mechanical properties of concrete, the predicted MOE in the models including density effect, are more compatible with the experimental results.

Keywords modulus of elasticity      normal strength normal weight concrete      empirical models      design codes      compressive strength      density     
Corresponding Authors: Behnam VAKHSHOURI   
Online First Date: 08 May 2018    Issue Date: 04 January 2019
 Cite this article:   
Behnam VAKHSHOURI,Shami NEJADI. Empirical models and design codes in prediction of modulus of elasticity of concrete[J]. Front. Struct. Civ. Eng., 2019, 13(1): 38-48.
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Fig.1  Stress-strain diagram of concrete and its components
Fig.2  Classification of factors affecting the elastic modulus of concrete [8]
Fig.3  Predicted MOE values of Eq. (1) versus experimented values
Fig.4  Comparison between predicted MOE in codes of practice vs. predicted MOE by proposed model.
Fig.5  Comparison between predicted MOE in empirical models vs. predicted MOE by proposed model.
Fig.6  Comparison of experimental vs. predicted MOE in proposed model and best matching models in codes of practice
Fig.7  Comparison of experimental vs. predicted MOE in proposed model and best matching empirical models
Design code Model Limits and coefficients
ACI 318- 08 [38] Ec=0.043w1.5(fc')0.5 MPa, kg/m3
Ec=4730(fc')0.5 MPa
1440<w<2500 kg/m3
Modified ACI 318-95
E in psi; fc' in psi; w in pcf
ACI 363R-92, (1997)[55] Ec=3320(fc')0.5+6890 21< fc'?<83 MPa
ACI-209-2R-08[56] Ec=0.043w1.5(fc')0.5 MPa, kg/m3
Ec=4730(fc')0.5 MPa
for 1440<w<2500 kg/m3
ACI 312-92 [8] E=14000+3250 fc'0.5
E=9500( fc'+8)1/3
E=0.043w1.5 fc'0.5 MPa
CSA A23.3-04[48] Ec=(3300 ( fc')0.5+6900)( w2300 ) 1.5 MPa
20≤ fc'≤40 MPa
1500<w<2500 kg/m3
CAN A23.3- M94[39] Ec=5(fc')0.5 GPa
NZS-3101-95[57] Ec=(3320 ( fc')0.5+6900)( w2300 ) 1.5MPa
CEB-FIP (1993) [44] Ec=10000 (fc'+8 ) 13
CEB-FIP (1990) [58] Ec=21500α( fc'10) 13 α=1.2 basalt, dense limestone,
1= quartzite , 0.9 limestone,
0.7 sandstone aggregate
EC2-04 (2004) [59] Ec=22000 ( fc' 10)0.3MPa sc between 0 and 0.4fcm
AASHTO-LRFD [45] Ec=0.043 k1 wc1.5(fc')0.5 kg/m3 and MPa
K1?= 1.0 unless determined by physical test
1280<w<2400 kg/m3
14≤ fc'≤48 MPa
FHWA (2000) [42] Ec=3837(fc')0.5 28≤ fc'≤193 MPa
OHBDC-1983 [46] Ec=5000(fc')0.5psi
NCHRP-2003 [43] Ec =33000k1k2 (0.14+fc'/1000) 1.5 ( fc ')0.5ksi k1 =1.0, k2 =90th percentile upper bound and the 10th percentile lower bound
2320<w<2480 kg/m3
AS-3600(2009) [51] Ec=0.043 wc1.5(fc')0.5
Ec=5050(fc')0.5normal weight concrete
<40 MPa, sc between 0 and 0.4fcm
AS-3600(2009) [51] Ec =wc1.5( 0.024 (fc') 0.5+0.12) fc'>40 MPa
JSCE (2007) [36] Ec=4700(fc')0.5(ACI-318-08)
Ec= 10.792ln( fc') - 9.0675 best fit
18≤ fc'≤80 MPa
NS-3473(1992) [60] Ec=9.5(fc')0.3( W2400)1.5 GPa, kg/m3
EHE (1998) [61] Ec=10,000fc'3
NBR-6118 (2003) [62] Ec=5600(fc')0.5
(AIJ- Japan) [47] Ec=2.1× 105 ( w2.3)1.5( fc'200)0.5
E, f’c: kgf/cm2 , w= t/m3
TS-500 (2000) [63] Ec=3.25(fc')0.5+14
IS 456 (BIS, 2000) [52] Ec=5000(fc')0.5
GBJ 11-89 (1994) [1] Ec= 102/[2.2 +( 34.7fc')]
IDC 3274 [1] Ec=5.7(fc')0.5
GDC 2000 [1] Ec=4.76(fc')0.5
SABS-0100 (1992)
Modified [4]
Ec (GPa) =K0+ afcu K0 (GPa) = 17 (ferro quartzite);
20 ( Jukskei granite);
29 (Eikenhof andesit
α (GPa/MPa) = 0.4 (ferro quartzite);
0.2 ( Jukskei granite and Eikenhof andesit
NTE E.060(2009) [49] Ec=0.043w1.5(fc')0.5 MPa, kg/m3
Ec=4730 ( fc')0.5 MPa
SP 52-101-2003 [14] Ec= 11.652ln(fc') - 7.4713 10≤ fc'≤60 MPa
BS 5400-4(1990) [37] Ec=8.6475 ( fc')0.348 20≤ fc'≤60 MPa
BS 8110 (1997)[50] Ec=20+0.2fc' 20≤ fc'≤60 MPa
Dutch VBC-95 [40] Ec=22250+250fc' MPa
RakMK-D3-2012 [53] Ec =5000(w2400) +(fc')0.5
Tab.1  Existing models in codes of practice to predict MOE of normal strength concrete
f’c (MPa) 10 15 20 25 30 35 40 45 50 55 60
E (GPa) 19 24 27.5 39 32.5 34.5 36 37 38 39 39.5
Best fit Ec= 11.652ln(fc') – 7.4713 R2= 0.9968 best fit
Tab.2  The best fitting equation with the CS-MOE data in SP 52-101(2003) [14]
f’c (MPa) 18 24 30 40 50 60 70 80
E (GPa) 22 25 28 31 33 35 37 38
Best fit Ec = 4700( fc'0.5 By ACI-318- R2=0.984
Ec = 10.792ln(f c') - 9.0675 Best fit R2 = 0.9983
Tab.3  The best fitting equation with the CS-MOE data in JSCE (2007) [36]
fc' (MPa) 20 25 30 35 40 60
E (GPa) 25 26 28 31 34 36
Best fit Ec=8.6475 ( fc')0.348, R2=0.992, best fit
Tab.4  The best fitting equation with the CS-MOE data in BS 5400-4(1990) [37]
Class-1 Class-2 Class-3 Class-4 Class-5
ACI-318-08[38] ACI312-92[8] ACI-363R-92[55] BS 8110 (1997)[50] CSA-A23.3-M94[39]
ACI-209-2R-08[56] TS-500-2000[63] NZS-3101-95[57] SABS 0100 (1992) Mdf.[4] OHBDC[46]
NTE-060-09[49] IS-456-2000[52]
JSCE-07[36] RakMK-D3-2012[53]
Tab.5  Classification of international design codes using the same MOE model
Researchers(s) Model Limits and coefficients
Carrasquillo, et al.(1981) [64] Ec=3.320 ( fc')0.5+6.900( w2346 ) GPa , kg/m3
Ec=3.320 ( fc')0.5+6.900
Dinakar (2008) [65] Ec=4.55(fc')0.5in fsp
Yanjun Liu (2006) [17] Ec=α ( fc')0.5
Ec , fc' in psi
α=55949 Miami oolite limestone, 62721 for Georgia granite ,
43777 for stalite lightweight aggregate
Rashid et al. (2002) [16] Ec=8900(fc')0.33 20< fc'<130 MPa
Kheder and Al-Windawi (2005) [6] Ec=5.323 ( fc')0.453 MPa, GPa
Soleymani (2006) [66] Ec=( wc2300)1.5( 3000 (fc') 0.5+6900)
Ec=( wc2300)1.5( 3000 (fc') 0.5+6900)
-Best fit with eq. 8-6 in CSA-A23.3(1994)
-Best fit with eq. 8-6 in CSA-23.3(1994)
-worst fit with eq. 8-7 in CSA-A23.3(1994)
Ravindrarajah et al. (1985) [67] Ec=4.630 ( fc')0.5
San Luis Obispo (2011) [68] Ec=6.59(fc')0.38
Noguchi et al. (2009) [3] MPa, kg/m3 40≤ fc'≤160 MPa
Haranki (2009) [5] Ec=31.92× w1.5(fc')0.5+345,328 psi , lb/ft3
Gardner and Zhao(1991) [69] Ec=9(fc')13 fc'>27 MPa
Ahmad and Shah (1985) [70] Ec=3.38× 10 5×λ2.5( fc')0.65
Jobse and Mustafa (1984) [71] Ec=0.103 wc1.5(fc')0.5
Cook (1989) [72] Ec=3.22× 10 5×λ2.5(fc')0.315 kg/m3 , MPa
Gutierrez and Canovas (1995) [73] Ec=8430fc'3
Leemann and Hoffmann (2005)[26] Ec=5480(fc')0.5
Min and Gjorv (1991) [7] Ec=1.19(fc') 2/3 GPa , MPa For light-weigth concrete
Levtchitch et el. (2004) [15] Ec=10000( 1.9+0.45fc') MPa
Jensen (1943) [52] Ec =6× 106/ (1+2000/fc' ) psi 1280<w<2400 kg/m3
14≤ fc'≤48 MPa
Pauw (1960) [41] Ec=13.82w1.79(fc')0.44
Tab.6  Empirical models to predict the modulus of elasticity of normal strength concrete
ACI-318-08[38] CSA.A23.3-M94[39] VBC-95[40] Proposed model
Predicted MOE
1.13 1.04 1.07 1.03
R2 0.73 0.63 0.71 0.78
Tab.7  Coefficient of correlation factor for MOE (models in codes of practice)
Pauw (1960)[41] Leemann and Hoffmann (2005)[26] Haranki (2009)[5] Proposed model
Predicted MOE
0.82 0.75 0.81 0.86
R2 0.99 0.97 1.02 1.003
Tab.8  Coefficient of correlation factor for MOE (Empirical models)
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